Properties

Label 4032.2.p.d
Level 4032
Weight 2
Character orbit 4032.p
Analytic conductor 32.196
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1344)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{2} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{2} ) q^{7} + ( -\beta_{1} - 2 \beta_{2} ) q^{11} + 4 q^{13} + \beta_{3} q^{17} -2 \beta_{1} q^{19} + 3 \beta_{1} q^{23} -5 q^{25} -\beta_{3} q^{29} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} + 2 \beta_{3} q^{37} + \beta_{3} q^{41} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{43} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{47} + ( 5 + \beta_{3} ) q^{49} -\beta_{3} q^{53} + 6 \beta_{1} q^{59} -4 q^{61} -3 \beta_{1} q^{71} + ( -12 - \beta_{3} ) q^{77} + \beta_{1} q^{79} -6 \beta_{1} q^{83} + \beta_{3} q^{89} + ( 4 \beta_{1} + 4 \beta_{2} ) q^{91} + 2 \beta_{3} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q + 16q^{13} - 20q^{25} + 20q^{49} - 16q^{61} - 48q^{77} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{2} \)\(/3\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - \nu^{2} + 3 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\((\)\( 2 \nu^{3} + 6 \nu \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + 2 \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\(3 \beta_{1}\)\(/2\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{3} - 6 \beta_{2} - 3 \beta_{1}\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.22474 + 1.22474i
−1.22474 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
0 0 0 0 0 −2.44949 1.00000i 0 0 0
1567.2 0 0 0 0 0 −2.44949 + 1.00000i 0 0 0
1567.3 0 0 0 0 0 2.44949 1.00000i 0 0 0
1567.4 0 0 0 0 0 2.44949 + 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.p.d 4
3.b odd 2 1 1344.2.p.b yes 4
4.b odd 2 1 inner 4032.2.p.d 4
7.b odd 2 1 4032.2.p.a 4
8.b even 2 1 4032.2.p.a 4
8.d odd 2 1 4032.2.p.a 4
12.b even 2 1 1344.2.p.b yes 4
21.c even 2 1 1344.2.p.a 4
24.f even 2 1 1344.2.p.a 4
24.h odd 2 1 1344.2.p.a 4
28.d even 2 1 4032.2.p.a 4
56.e even 2 1 inner 4032.2.p.d 4
56.h odd 2 1 inner 4032.2.p.d 4
84.h odd 2 1 1344.2.p.a 4
168.e odd 2 1 1344.2.p.b yes 4
168.i even 2 1 1344.2.p.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.a 4 21.c even 2 1
1344.2.p.a 4 24.f even 2 1
1344.2.p.a 4 24.h odd 2 1
1344.2.p.a 4 84.h odd 2 1
1344.2.p.b yes 4 3.b odd 2 1
1344.2.p.b yes 4 12.b even 2 1
1344.2.p.b yes 4 168.e odd 2 1
1344.2.p.b yes 4 168.i even 2 1
4032.2.p.a 4 7.b odd 2 1
4032.2.p.a 4 8.b even 2 1
4032.2.p.a 4 8.d odd 2 1
4032.2.p.a 4 28.d even 2 1
4032.2.p.d 4 1.a even 1 1 trivial
4032.2.p.d 4 4.b odd 2 1 inner
4032.2.p.d 4 56.e even 2 1 inner
4032.2.p.d 4 56.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5} \)
\( T_{11}^{2} - 24 \)
\( T_{13} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ \( ( 1 + 5 T^{2} )^{4} \)
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{4} \)
$17$ \( ( 1 - 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 22 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 34 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 + 38 T^{2} + 961 T^{4} )^{2} \)
$37$ \( ( 1 + 22 T^{2} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 58 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 10 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( ( 1 - 2 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 82 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 26 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 4 T + 61 T^{2} )^{4} \)
$67$ \( ( 1 + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 106 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 73 T^{2} )^{4} \)
$79$ \( ( 1 - 154 T^{2} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 22 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 154 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 98 T^{2} + 9409 T^{4} )^{2} \)
show more
show less